It seems clear that the quantum curvature is connected to the presence of entanglement, with the trivial phases appearing completely flat. A phase with no entanglement would then be expected to have zero curvature, except perhaps when the critical point is approached. But it is not clear whether or not the presence of quantum curvature is only incidental in the Haldane phase, and is in fact due to the nearby quantum critical points which exhibit a massive divergence in the amount of GME.
This would account for the presence of small amounts of positive curvature in the Néel and insulator phases as the critical points are approached. In this case the positive curvature in the Haldane phase is a consequence of the phases existing in between these two critical points. This introduces the possibility of detecting the presence of quantum critical points even outside of the scaling regime, without the need to directly witness the diverging multipartite entanglement at the critical point. This would be especially useful in inelastic neutron scattering experiments where resolution issues make it effectively impossible to directly measure diverging multipartite entanglement [1]}.
That state space curvature is induced by the phase boundary implies that quantum critical regions have long range effects in the state space, influencing not just the geometry of states in the scaling regime, but also the states that lie in between these critical points.
